Question

Multiply the rational expression and write the product in simplest form: x2+2x−3x2+3x−4⋅x2+6x+8x2−9.(1 point) Responses −23 negative Start Fraction 2 over 3 End Fraction x+2x−3 Start Fraction x plus 2 over x minus 3 End Fraction −(x+2)x+3 Start Fraction negative left parenthesis x plus 2 right parenthesis over x plus 3 End Fraction (x+4)(x+2)(x+3)(x−3)

Answer 1

To multiply the rational expressions \(\frac{x^2 + 2x - 3}{x^2 + 3x - 4} \cdot \frac{x^2 + 6x + 8}{x^2 - 9}\) and simplify, let's start by factoring each polynomial.

1. **Factor \(x^2 + 2x - 3\)**:
This factors as \((x + 3)(x - 1)\).

2. **Factor \(x^2 + 3x - 4\)**:
This factors as \((x + 4)(x - 1)\).

3. **Factor \(x^2 + 6x + 8\)**:
This factors as \((x + 2)(x + 4)\).

4. **Factor \(x^2 - 9\)**:
This is a difference of squares and factors as \((x + 3)(x - 3)\).

Now we can rewrite the expression with these factored forms:

\[
\frac{(x + 3)(x - 1)}{(x + 4)(x - 1)} \cdot \frac{(x + 2)(x + 4)}{(x + 3)(x - 3)}
\]

Next, we multiply the numerators and the denominators:

Numerator:

\[
(x + 3)(x - 1)(x + 2)(x + 4)
\]

Denominator:

\[
(x + 4)(x - 1)(x + 3)(x - 3)
\]

Now we combine the fractions:

\[
\frac{(x + 3)(x - 1)(x + 2)(x + 4)}{(x + 4)(x - 1)(x + 3)(x - 3)}
\]

Now we can cancel out the common factors in the numerator and the denominator:

- \((x + 3)\)
- \((x - 1)\)
- \((x + 4)\)

This leaves us with:

\[
\frac{(x + 2)}{(x - 3)}
\]

Thus, the product in simplest form is:

\[
\frac{x + 2}{x - 3}
\]

So, the final answer is:

**\(\frac{x + 2}{x - 3}\)**